The authors establish some new sufficient conditions under which all solutions of a certain class of nonlinear neutral delay differential equations of the third order are stable, bounded, and square integrable. Illustrative examples are given to demonstrate the main results.

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)+b\left(t\right)x\left(h\left(t\right)\right)=0,$$ where $$\left|a\right(t\left)\right|<1,b\left(t\right)\ge 0,h\left(t\right)\le t,g\left(t\right)\le t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation $${\left[a\left(t\right){\left({\left[x\left(t\right)+p\left(t\right)x\left(\sigma \left(t\right)\right)\right]}^{\text{'}\text{'}}\right)}^{\gamma }\right]}^{\text{'}}+q\left(t\right)f\left(x\left[\tau \left(t\right)\right]\right)=0,t⩾t_0.\left(E\right)$$ . We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.

Autonomous linear neutral delay and, especially, (non-neutral) delay difference equations with continuous variable are considered, and some new results on the behavior of the solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha }\right]}^{\text{'}}+q\left(t\right)x^{\alpha }\left(\tau \left(t\right)\right)=0,E$$ where $\alpha >0$, $0\le p\left(t\right)\le p_0<\infty$ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

The problems of both single and multiple delays for neutral-type uncertain systems are considered. First, for single neutral time delay systems, based on a Razumikhin-type theorem, some delay-dependent stability criteria are derived in terms of the Lyapunov equation for various classes of model transformation and decomposition techniques. Second, robust control for neutral systems with multiple time delays is considered. Finally, we demonstrate numerical examples to illustrate the effectiveness...

The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation $${(x\left(t\right)-px(t-\tau ))}^{\text{'}\text{'}}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0$$ to be oscillatory and to improve some existing results. The main results are based on the comparison principles.